Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls. So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
If you added the modifier to the bonus damage, the spell would specify that. So if you are level 5 and use booming blade with a rapier (and hit), the damage would be 1d8+1d8+str/dex bonus and an additional 2d8 if the target willingly moves on their turn.
Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls. So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
That 1d8 damage would not include your Str modifier for a non-finesse weapon, so it does not include your Dex modifier for a finesse weapon.
It would be kind of broken of you could add your ability mod to it because you make a weapon attack witch already adds your mod to the weapon damage, even with point buy at the lowest level that is like a extra 6 damage on rolls
Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls. So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
The closest that you can come to double dipping on modifiers and using Booming Blade would be with:
Arcana Cleric 8- You'll get one with your melee attack stat and one with Wisdom from potent spellcasting. The only way it actually becomes double dipping is if you somehow get Shillelagh through multiclass, magic initiate, wood elf magic, or some other means. You have to take Booming Blade as one of the wizard cantrips that count as cleric cantrips.
Evocation Wizard 10- Empowered Evocation lets you add your Intelligence modifier to any evocation spell, which Booming Blade is. You're still using your melee attack stat.
Bladesinger Wizard 14- Song of Victory Starting at 14th level, you add your Intelligence modifier (minimum of +1) to the damage of your melee weapon attacks while your Bladesong is active. Technically, Booming Blade uses a melee weapon attack as part of the spell. YMMV as it uses melee attack with a weapon as the verbiage, which may lead a DM to disallow it. But it is RAW.
For either of the Wizard options, you could go Battle Smith Artificer 3 for Battle Ready, the second bullet reads as follows: When you attack with a magic weapon, you can use your Intelligence modifier, instead of Strength or Dexterity modifier, for the attack and damage rolls. That would allow you to double dip with intelligence under specific circumstances and assuming that your DM doesn't read attack with a capital A. It is noteworthy, that casting Booming Blade wouldn't qualify it as a magic attack since the melee attack with a weapon doesn't make the weapon damage magical. Artificer doesn't have a way to add modifier damage otherwise that I saw.
Perhaps something similar from a class I'm not familiar with. Additionally, the following might be possible, but they aren't precisely RAW. Stop reading if that's all you're interested in, unless you want to read the reasons why they wouldn't work.
Eldritch Knight 7-War Magic gives you an extra weapon attack as a bonus action after casting a cantrip. It says nothing about adding a damage modifier and therefore isn't eligible RAW. If you can talk your DM into Rule of Cool, it could be added via some homebrew or another. It could at least make sense. I'm blanking atm on whether Extra Attack even lets you double dip, so maybe not.
Draconic Bloodline Sorcerer 6- Elemental Affinity reads: Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell. The only problem is that Thunder damage isn't an eligible draconic ancestry choice and therefore isn't eligible RAW. Your DM might be willing to fudge it for lightning, though.
Notably, Storm Sorcery doesn't offer an option because the closest option (Heart of the Storm) specifies a 1st level or higher spell.
I certainly didn't exhaust my resources trying to find options, so there may be more. Booming Blade is only a Sorcerer, Warlock, or Wizard spell, and most abilities that let you add your modifier to spells at all usually require it be a class spell to apply. Warlock doesn't give the option for their cantrips (and probably spells) outside of Agonizing Blast, which isn't applicable. One of the Bard subclasses might, but it didn't look like it from the PHB ones, but Magical Secrets (Bard 10) could be used to get Booming Blade as a bard spell if one of the subclasses could.
In short, there aren't a lot of options for double dipping your modifier bonuses, particularly a single modifier. Those options are typically gated behind higher level abilities and usually from spellcaster focused subclasses. You can manage a build, but it'll probably be awkward.
Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls. So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
The answer is in your question, you add the modifier to the damage roll not the damage dice so in this case
Assuming you mean the secondary bonus damage from the target moving, no. That's not weapon damage, and by the time it triggers, your attack - and all resulting weapon damage - will have already resolved.
Double dipping bonuses, as other posters have suggested, is a different matter. It pretty much always involves multiclassing and tends to be a high-level thing, with the sole exception of Green Flame Blade Sorlocks, who can gain double modifier damage as early as seventh level. Very high level Hexblade/Fire Dragon Warcerers can triple dip Charisma on GFB - once with the basic weapon attack, twice from Elemental Affinity (Fire), and a third time from the Lifedrinker Invocation at 12th level. Requires a minimum character level of 18, but if you go with Great Weapon Master and a giant sword, you can start hitting 25 flat damage per GFB, before you roll.
Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls. So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
No.
The Finesse property is only applicable to damage from weapon attacks. Booming Blade is a spell that involves a weapon attack. You add your Str/Dex modifier only to that portion of the attack, and that aspect never changes.
At character level 5, the spell deals an additional 1d8 flat Thunder damage--no modifier is added to this. The secondary Thunder damage from the target moving does not get a modifier added either.
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Finesse weapons allow you to add str/dex modifier to attack and all of the damage rolls.
So when I use booming blade on level 5, can I add that modifier to that 1d8 bonus damage?
If you added the modifier to the bonus damage, the spell would specify that. So if you are level 5 and use booming blade with a rapier (and hit), the damage would be 1d8+1d8+str/dex bonus and an additional 2d8 if the target willingly moves on their turn.
That 1d8 damage would not include your Str modifier for a non-finesse weapon, so it does not include your Dex modifier for a finesse weapon.
It would be kind of broken of you could add your ability mod to it because you make a weapon attack witch already adds your mod to the weapon damage, even with point buy at the lowest level that is like a extra 6 damage on rolls
The closest that you can come to double dipping on modifiers and using Booming Blade would be with:
Arcana Cleric 8- You'll get one with your melee attack stat and one with Wisdom from potent spellcasting. The only way it actually becomes double dipping is if you somehow get Shillelagh through multiclass, magic initiate, wood elf magic, or some other means. You have to take Booming Blade as one of the wizard cantrips that count as cleric cantrips.
Evocation Wizard 10- Empowered Evocation lets you add your Intelligence modifier to any evocation spell, which Booming Blade is. You're still using your melee attack stat.
Bladesinger Wizard 14- Song of Victory Starting at 14th level, you add your Intelligence modifier (minimum of +1) to the damage of your melee weapon attacks while your Bladesong is active. Technically, Booming Blade uses a melee weapon attack as part of the spell. YMMV as it uses melee attack with a weapon as the verbiage, which may lead a DM to disallow it. But it is RAW.
For either of the Wizard options, you could go Battle Smith Artificer 3 for Battle Ready, the second bullet reads as follows: When you attack with a magic weapon, you can use your Intelligence modifier, instead of Strength or Dexterity modifier, for the attack and damage rolls. That would allow you to double dip with intelligence under specific circumstances and assuming that your DM doesn't read attack with a capital A. It is noteworthy, that casting Booming Blade wouldn't qualify it as a magic attack since the melee attack with a weapon doesn't make the weapon damage magical. Artificer doesn't have a way to add modifier damage otherwise that I saw.
Perhaps something similar from a class I'm not familiar with. Additionally, the following might be possible, but they aren't precisely RAW. Stop reading if that's all you're interested in, unless you want to read the reasons why they wouldn't work.
Eldritch Knight 7-War Magic gives you an extra weapon attack as a bonus action after casting a cantrip. It says nothing about adding a damage modifier and therefore isn't eligible RAW. If you can talk your DM into Rule of Cool, it could be added via some homebrew or another. It could at least make sense. I'm blanking atm on whether Extra Attack even lets you double dip, so maybe not.
Draconic Bloodline Sorcerer 6- Elemental Affinity reads: Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell. The only problem is that Thunder damage isn't an eligible draconic ancestry choice and therefore isn't eligible RAW. Your DM might be willing to fudge it for lightning, though.
Notably, Storm Sorcery doesn't offer an option because the closest option (Heart of the Storm) specifies a 1st level or higher spell.
I certainly didn't exhaust my resources trying to find options, so there may be more. Booming Blade is only a Sorcerer, Warlock, or Wizard spell, and most abilities that let you add your modifier to spells at all usually require it be a class spell to apply. Warlock doesn't give the option for their cantrips (and probably spells) outside of Agonizing Blast, which isn't applicable. One of the Bard subclasses might, but it didn't look like it from the PHB ones, but Magical Secrets (Bard 10) could be used to get Booming Blade as a bard spell if one of the subclasses could.
In short, there aren't a lot of options for double dipping your modifier bonuses, particularly a single modifier. Those options are typically gated behind higher level abilities and usually from spellcaster focused subclasses. You can manage a build, but it'll probably be awkward.
The answer is in your question, you add the modifier to the damage roll not the damage dice so in this case
damage roll = booming blade damage dice + weapon damage dice + str/dex modifier
Assuming you mean the secondary bonus damage from the target moving, no. That's not weapon damage, and by the time it triggers, your attack - and all resulting weapon damage - will have already resolved.
Double dipping bonuses, as other posters have suggested, is a different matter. It pretty much always involves multiclassing and tends to be a high-level thing, with the sole exception of Green Flame Blade Sorlocks, who can gain double modifier damage as early as seventh level. Very high level Hexblade/Fire Dragon Warcerers can triple dip Charisma on GFB - once with the basic weapon attack, twice from Elemental Affinity (Fire), and a third time from the Lifedrinker Invocation at 12th level. Requires a minimum character level of 18, but if you go with Great Weapon Master and a giant sword, you can start hitting 25 flat damage per GFB, before you roll.
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No.
The Finesse property is only applicable to damage from weapon attacks. Booming Blade is a spell that involves a weapon attack. You add your Str/Dex modifier only to that portion of the attack, and that aspect never changes.
At character level 5, the spell deals an additional 1d8 flat Thunder damage--no modifier is added to this. The secondary Thunder damage from the target moving does not get a modifier added either.
You don't know what fear is until you've witnessed a drunk bird divebombing you while carrying a screaming Kobold throwing fire anywhere and everywhere.